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CENTER OF MASS AND MOMENT OF INERTIA     157
2. Find the moment of inertia of a cylinder about a transverse axis through the center of mass.
Let m, a, I, and r be, respectively, the mass, the radius, the length, and the density of the cylinder.   Further let the given axis pass through the center of mass of the cylinder; then taking a slice obtained by two right sections as the element of mass we get, by theorem II, dly = dly' + z1 dm,
where dm is the mass of the element, dly and dly> are the moments of inertia of the element about the given axis and about a parallel axis through the center of mass of the element, and z is the distance between these two axes. But by theorem I
dls + dl* = dl,,
and by symmetry                    dlj = dly>,
and by the last illustrative example
a2 dm
dlm
Therefore
2
a2 dm
Substituting this value of dly in the expression for dly we get dly
Integrating the last equation we have
_ma?  .   r *    ____